IB Mathematics Applications & Interpretation Topic…

A model for the spread of a rumor involves susceptible ( $x$ ) and informed ( $y$ ) individuals:

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Show that the system has a line of equilibrium points on the $x$ -axis (i.e. $y=0$ ).

For the equilibrium point $(0,0)$ , find the Jacobian matrix and its eigenvalues, and classify the stability.

Given initial conditions $x(0)=8, y(0)=1$ , use Euler's method with a step size of $0.1$ days to the populations after $0.3$ days.

the phase portrait on $0\le x\le 10$ , $0\le y\le 3$ . Show the equilibrium line(s) and at least three trajectories with direction arrows.

A model for the populations of two symbiotic algae species, $x$ and $y$ (in thousands per $\mathrm{m}^{2}$ ), is given by:

\frac{d x}{d t}=x+2 y-10, \quad \frac{d y}{d t}=2 x+y-15

Time $t$ is measured in arbitrary time units.

Find the equilibrium point.

Use the substitution $X=x-\frac{20}{3}, Y=y-\frac{5}{3}$ to transform the system into a homogeneous system, and find its eigenvalues.

The displacement of a damped spring-mounted platform, $q$ (in $\mathrm{cm}$ ), is modeled by

\frac{d^{2} q}{d t^{2}}+2 \frac{d q}{d t}+5 q=15

where $t$ is in seconds.

Convert the equation to a system of first-order differential equations using $y=\frac{d q}{d t}$ , and find the equilibrium point.

A predator-prey system models the populations of deer ( $D$ , in thousands) and wolves ( $W$ , in hundreds) in a national park, given by:

\frac{\mathrm{d}D}{\mathrm{d}t} = 0.45D - 0.03DW, \quad \frac{\mathrm{d}W}{\mathrm{d}t} = 0.02DW - 0.4W

where $t$ is in years. Initially, $D = 22, W = 10$ .

Find the non-zero equilibrium point for the system.

the equilibrium point in the context of the model.

Use Euler's method with a step size of $0.2$ years to $D$ and after years.

For real $a$ , $M_a=\begin{pmatrix}a&2\\2&a\end{pmatrix}$ . Values giving a zero eigenvalue are excluded by the model.

Find the two eigenvalues and their fixed eigendirections in terms of $a$ .

Find the complete set of $a$ for which the origin is a saddle, enforcing the non-zero-eigenvalue restriction.

Classify the origin for the remaining allowed ranges and what happens at the two excluded boundary values.

Find the general solution of the original system, given the eigenvalues and eigenvectors from Part 1.

Given initial conditions $x(0)=5, y(0)=3$ , find the particular solution and the limit of $\frac{y}{x}$ as $t \rightarrow \infty$ .

the phase portrait for $0 \leq x \leq 6,0 \leq y \leq 6$ , showing the equilibrium point and trajectories.

Find the eigenvalues of the homogeneous system's matrix.

Find the exact solution for $q(t)$ , given $q(0)=0,\; y(0)=0$ .

the graph of $q$ against $t$ for $0 \leq t \leq 5$ , labeling the equilibrium level.

On a phase plane with $D$ on the horizontal axis and $W$ on the vertical axis, for $18 \leq D \leq 24$ and $8 \leq W \leq 12$ , the initial direction of the trajectory from $(22,10)$ .

IB Mathematics Applications & Interpretation AHL 5.17—phase Portrait Questionbank

IB Mathematics Applications & Interpretation AHL 5.17—phase Portrait Questionbank

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IB Mathematics Applications & Interpretation AHL 5.17—phase Portrait Questionbank

IB Mathematics Applications & Interpretation AHL 5.17—phase Portrait Questionbank

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IB Mathematics Applications & Interpretation (AI) Topic AHL 5.17—phase Portrait Questionbank